My Math Forum Ring Homo.

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 December 12th, 2009, 06:40 AM #1 Member   Joined: Oct 2009 Posts: 85 Thanks: 0 Ring Homo. $f(z)= x^3$ 1) show f not a ring homo. over $\mathbb{Z}$ 2) show f is a ring homo. over $\mathbb{Z}/3\mathbb{Z}$ for part 2) we have $x= 0, 1, 2$. Then $f(x)= x^3 = x(mod3)$. So $\forall$ $x,y \in \mathbb{Z}/3\mathbb{Z}$ we have $f(x+y)= x+y = f(x) + f(y)$ and $f(xy)= xy = f(x) f(y)$. Is this all that has to be shown for 2? And for part one I dont know how to go about not showing homo.
 December 12th, 2009, 11:55 AM #2 Senior Member   Joined: Jan 2009 From: Japan Posts: 192 Thanks: 0 Re: Ring Homo. For part 1, just show any counterexample to f(x + y) = f(x) + f(y) or f(xy) = f(x)f(y). I recommend the additive portion -- every counterexample I can think of for the multiplicative portion exploits the problems of the additive one.

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